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The Joint Approach

For the joint approach, the reduction proceeds in a fashion which appears very similar to conventional observations. First a dirty image is formed (with associated point-spread function), and then a deconvolution algorithm is used to `clean' this dirty image. Finally the `restore' step is performed. There are, however, substantial differences - although these are largely hidden from the user.

The task to form the dirty image is still invert. The dirty image is formed by imaging (using a conventional algorithm) each of the pointings separately. These individual pointing images are them combined in a linear mosaicing process. This linear mosaicing simply consists of a weighted average of the pixels in the individual pointings, with the weights determined by the primary beam response and the expected noise level. The resultant output dirty image is thus an image of the entire region mosaiced.

The weights are computed to minimise the noise in the resultant image as well as to correct for primary beam attenuation. The output image, I(l,m), is given by

$$I(\ell,m) = g(\ell,m) \frac{\sum_i P(\ell-\ell_i,m-m_i)I_i(\ell,m)/\sigma_i^2} {\sum_i P^2(\ell-\ell_i,m-m_i)/\sigma_i^2}.$$

Here the summation, i, is over the set of pointing centres, $(\ell_i,m_i)$. $I_i(\ell,m)$ is the image formed from the i'th pointing, and P(l,m) is the primary beam pattern. The expected noise variance in the i'th field is $\sigma_i^2$.

Primary beam attenuation is only corrected for within limits. Because there are large variations in sensitivity across a mosaiced image (the edges of a mosaiced region will have low sensitivity), the imaging software does not always attempt to fully correct for primary beam attenuation. Instead, it constrains the weights so that the noise level does not exceed a certain limit (this limit is based on the noise in individual pointings). This results in some residual primary beam attenuation at the edges of a mosaic (or in holes in the pointing grid). This is done by the function g(l,m). This function normally has a value of 1, but its value drops towards 0 at edges or holes. In this way, the noise level across a mosaiced image is crudely uniform.

Task invert also applies geometric corrections to account for the fact that the sky is not a plane. For an east-west array, such as the ATCA, these corrections are exact, meaning that the coordinate geometry of the resultant images is also (nominally) exact. For other array types, the corrections are optimal in the sense that they are the best approximation that still results in a convolution relationship (in the sense that such arrays obey a convolution relationship!).

Because the u-v coverage of the different pointings will not be identical, the synthesised beam patterns will differ between pointings. This, and the weighted average process, means that the point-spread function of the resultant dirty image is position-dependent. As most deconvolution algorithms assume a position-independent point-spread function, a conventional algorithm cannot be used. However the point-spread function from the linear mosaicing process is still reasonably compactly described and readily computed. The beam dataset that invert produces is not a normal one; it is a cube of beam patterns, one for each pointing. Given this, and some information stored in an auxiliary mosaic information table, the deconvolution tasks can compute the true point-spread function at any position in the dirty image. Being able to compute a point-spread function (or rather, being able to compute a dirty image, given a trial deconvolved image) is the difficult part of writing a deconvolution task. A maximum entropy-based deconvolution algorithm is readily implemented.

The practicalities of this processing are now described.